I hope this is not off topic, but I'm working with a process $(X_n, n\in \mathbb{N})$ and there's a theorem I want to use that is valid for certain strong mixing processes.
The thing is, the usual definitions of mixing processes usually start off with a `let $(X_n, n\in \mathbb{Z})$...' etc. Is there any particular reason to require that the index set be $\mathbb{Z}$ in the definition?
For instance, strong mixing is defined in the following manner: first we set $\alpha(\mathscr{A},\mathscr{B}):=\sup \{\vert P(A\cap B)-P(A)P(B)\vert\; :\;A\in\mathscr{A},B\in\mathscr{B}\}$ for sub-$\sigma$-algebras $\mathscr{A},\mathscr{B}$ of $\mathscr{F}$. Then we define, for $J\in\mathbb{Z}$, the $\sigma$-algebras $\mathscr{F}^J:=\sigma(X_j,\;j\leq J)$ and $\mathscr{F}_J:=\sigma(X_j,\;j\geq J)$. Then we set, for $n\in\mathbb{N}$, $\tilde{\alpha}(n):=\sup_{J\in\mathbb{Z}}\alpha(\mathscr{F}^J,\mathscr{F}_{J+n})$ and say that $(X_n)$ is strongly mixing iff $\tilde{\alpha}(n)\rightarrow 0$.
So here's my question: can we give a similar definition of strongly mixing for a process $(X_n,\;n\in\mathbb{N})$?